Root Locus Analysis (2)

1. Root Locus Analysis (2) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University

2. General Overview • This section explain the root locus plot for positive feedback system instead of negative feedback • All rules use the opposite of the ones used in the negative feedback

3. R(s) C(s) G(s) + + H(s) Standardization 1 – G(s)H(s) = 0 Find Characteristic Equation!!

4. How to make it? • Start from the characteristic equation • Locate the poles and zeros on the s plane • Determine the root loci on the real axis • Determine the asymptotes of the root loci • Find the breakaway and break-in points • Determines the angle of departure (angle of arrival) from complex poles (zeros) • Find the points where the root loci may cross the imaginary axis

5. Example (1) 1. Start from the characteristic equation

6. Example (2) 2. Locate the poles and zeros on the s plane

7. Example (3) 3. Determine the root loci on the real axis

8. Example (4a) 4. Determine the asymptotes of the root loci

9. Example (4b) 4. Determine the asymptotes of the root loci

10. Example (5) From the characteristic equation, find then calculate… s = -0.8 and s = -2.35±0.77j 5. Find the breakaway and break-in point

11. Example (6) qpole= 0 – sum from pole + sum from zero qzero = 0 – sum from zero + sum from pole qpole= 0 – 27 – 90 + 45 = -72 6. Determine the angle from complex pole/zero

12. Example (7) Do this part by substituting jw for all s in the characteristic equation K = - w2+8 and - w2 = -10/3 The root locus does not cross the jw axis 7. Find the points where the root loci may cross the imaginary axis

13. Example (8)

14. Plot in negative feedback

15. Root Locus in Matlab Function rlocus(num,den) draws the Root Locus of a system. Another version in state space is rlocus(A,B,C,D) The characteristic equation For positive feedback use rlocus(-num,den) to plot the Root Locus in complex plane

16. Next… Design of a compensator is the topic for the next meeting… See you there!